1. NSF DUE-0126716 Collaborative Research: Adaptation & Implementation of Activity & Web-Based Materials Into Post-Calculus Introductory Probability and Statistics Courses Tracy Goodson-Espy M. Leigh Lunsford Ginger Holmes Rowell

2. NSF DUE-0126716 A Collaborative Approach Athens State Univ. M. Leigh Lunsford Middle Tenn. St. Univ. Ginger Holmes Rowell Univ. of Alabama, Huntsville Tracy Goodson-Espy A&I Materials into Post Calculus Prob/Stat Courses Provide Objective Independent Assessment of A&I

3. NSF DUE-0126716 Project* Objectives To improve post-calculus students learning of probability & statistics. To provide students with better preparation for their future careers in mathematics & statistics, mathematics education, and computer science. *This project is partially supported by the National Science Foundation. The project started in June, 2002 and continues through August, 2004.

4. NSF DUE-0126716 Courses for A&I ASU: Applied Statistics & Probability I (3 hrs) Clientele: CS, MA, Math. Ed. Majors Prereq: Calculus II MTSU: Probability & Statistics (3 hrs) Data Analysis (1 hr) Clientele: CS, Math. Ed. Majors Prereq: Calculus I

5. NSF DUE-0126716 The Materials for A&I “A Data-Oriented, Active Learning, Post- Calculus Introduction to Statistical Concepts, Methods, and Theory (SCMT)”“A Data-Oriented, Active Learning, Post- Calculus Introduction to Statistical Concepts, Methods, and Theory (SCMT)” A. Rossman, B. Chance, K. Ballman NSF DUE-9950476 “Virtual Laboratories in Probability and Statistics (VLPS)”“Virtual Laboratories in Probability and Statistics (VLPS)” K. Siegrist NSF DUE-9652870

6. NSF DUE-0126716 Statistical Concepts, Methods, and Theory (SCMT): A Small Sample of Materials ActivityContextConcepts Description Random- ization Test Friendly Observers Randomization, simulation, p-value Uses cards (23 per student) and Minitab to simulate a randomization test to estimate p- value from a 2x2 table for a psychology study. Equal Likeliness Random Babies Sample space, long-run relative frequency, random variable, expected value, simulation Uses index cards (4 per student) and Minitab to simulate the matching problem and develops probability calculations with equally likely outcomes. Fishers Exact Test Friendly Observers Counting rules, hypergeometric probabilities Exact probabilities for simulation in Randomization Test General vs. Specific The Birthday Problem Applications of counting techniques, complement rule Does calculations for the birthday problem (using a spreadsheet) contrasting any birthday vs. a specific birthday Prob. Rules 100 top films, 2000 Michigan primary Variety of basic probability rules Discovery approach through two-way tables and some Venn diagrams. HW is very interesting

7. NSF DUE-0126716 Virtual Laboratories in Probability & Statistics: An Example Games of Chance Contents 1.PokerPoker 2.Poker Dice and Chuck-a-LuckPoker Dice and Chuck-a-Luck 3.CrapsCraps 4.IntroductionIntroduction 5.RouletteRoulette 6.The Monty Hall ProblemThe Monty Hall Problem 7.LotteriesLotteries 8.NotesNotes Applets Poker Experiment Poker Dice Experiment Chuck-a-Luck Experiment Craps Experiment Roulette Experiment Monty Hall Game Monty Hall Experiment Poker Experiment Applet

8. NSF DUE-0126716 Topics in ASU’s Course Topics Included: Random Experiments, Sample Spaces, Random Samples Basic Descriptive Statistics (Mean, Var., Std. Dev., Sample Mean and Var.) Probability Theory: Laws of Probability, Conditional Probability, Independence, Law of Total Probability, Bayes’ Theorem Discrete and Continuous Random Variables Uniform (discrete & continuous), Binomial, Hypergeometric, Geometric, Normal, pdf vs. cdf. Expected Value Central Limit Theorem Confidence Intervals for Means & Proportions Basic Concepts in Hypothesis Testing Topics Not Included: Multivariate Distributions Moment Generating Functions NOTE: Topics not covered linearly!

9. NSF DUE-0126716 An Example Friendly Observers Experiment*: Researchers investigated a conjecture that having an observer with a vested interest would decrease subjects’ performance on a skill-based task. Subjects given time to practice task. Subjects randomly assigned to one of two groups: Group (A) was told that the participant and observer would each win $3 if the participant beat a certain threshold time. Group (B) was told only that the participant would win $3 if the threshold were beaten. Threshold chosen to be a time that participant beat in 30% of their practice turns. A: observer shares prizeB: no sharing of prizeTotal Beat threshold38 11 Do not beat threshold94 13 Total 12 12 24 *Journal of Personality and Social Psychology (Butler and Baumeister, 1998)

10. NSF DUE-0126716 An Example Week 1: Activity with the Friendly Observers Experiment Randomness, random variable, empirical probability distribution, p-value. (Hypothesis testing) Uses a tactile simulation (in-class) and Minitab (assignment) to determine the empirical probability distribution for the random variable X (the number of winners assigned to group A by chance). Students start writing a report detailing their simulation results and their empirical estimate of the p-value. Week 3: Follow-up activity includes Friendly Observers Exper. Introduces hypergeometic probabilities and the hypergeometric distribution. Students apply theory to Friendly Observers Experiment above to compute the probability distribution of X and the actual p-value of the experiment. Uses the Ball and Urn Applet from the Virtual Labs. Students finish report.

11. NSF DUE-0126716 Theoretical Orientation The project is based on an emergent constructivist perspective, * meaning that mathematics learning can be characterized as both a process of active individual construction and a process of enculturation. This orientation emphasizes the importance of analyzing students’ individual mathematical activities as well as placing them in the context of the mathematical community in which they were developed. This is done through the use of a teaching experiment. Students’ individual constructive activities and their roles in social processes in the classroom act in a complementary way to enable student learning. *1995, P. Cobb

12. NSF DUE-0126716 Teaching Experiment Cycle Teaching Hypotheses; Curricular & Instructional Choices Class Implementation & Feedback Instructors’ Reflections and Curricular Modifications

13. NSF DUE-0126716 The A&I Cycle SCMT VLPS Materials Use Materials in Courses Evaluate A&I Refine Adaptation Provide Feedback to Material Developers Use Independent Assessment Results to Improve A&I Develop Best Practices for Use of Materials Co-jointly Receive Training on New Materials

14. NSF DUE-0126716 Assessment of A&I of Materials Action Research ModelWill Use an Action Research Model* –What is the problem? I.e., what is not working in the classroom? –What technique can be used to address the learning problem? –What type of evidence can be gathered to show whether the implementation is effective? –What should be done next, based on what was learned? *1999 - R. delMas, J. Garfield, B. Chance

15. NSF DUE-0126716 Project Goals and Outcomes The development of post-calculus probability and statistics courses that produce well-educated students. The integration of technology and group-based activity work into the courses for the purpose of enhancing student learning. The enhancement of student communication skills through oral and written reports and presentations. The improvement and implementation of non-traditional assessment techniques for evaluating students. A contribution to the mathematics community discussion/research concerning what topics/materials/methods should be included in reform- oriented probability and statistics courses to improve overall student understanding of the subject.

16. NSF DUE-0126716 Project Goals and Outcomes Goal: The development of post-calculus probability and statistics courses that produce well-educated students. Outcome: The project uses SCMT and VLPS for Three Different Types of Courses: Athens State University - Mathematics 331 Applied Statistics & Probability (primarily an undergraduate probability course) Middle Tennessee State University - Math 2050/Statistics 5140 Probability & Statistics (primarily an undergraduate course in inferential statistics) Middle Tennessee State University- Mathematics 6350 Probability and Statistics for Teachers

17. NSF DUE-0126716 Project Goals and Outcomes Goal: The integration of technology and group-based activity work into the courses for the purpose of enhancing student learning. Outcome: The project integrates selected SCMT and VLPS materials into the courses as well as using Mini- tab. Using these materials, students work in groups on discovery-based learning activities. The technology is integrated for three purposes 1) a tool to help with computations, 2) a tool for visualization and 3) a tool for simulation.

18. NSF DUE-0126716 Project Goals and Outcomes Goal: The enhancement of student communication skills through oral and written reports and presentations. Outcome: Each of the courses included in the project requires multiple oral and written student reports and presentations that explore student understandings of the concepts covered in class and through the individual and group activities.

19. NSF DUE-0126716 Project Goals and Outcomes Goal: The improvement and implementation of non- traditional assessment techniques for evaluating students. Outcome: Each of the courses included in the project uses assessment methods in addition to traditional paper and pencil tests. Non-traditional assessments include written and oral reports, homework including the analysis of real data, written artifacts from lab activities and group work.

20. NSF DUE-0126716 Project Goals and Outcomes Goal: A contribution to the mathematics community discussion/research concerning what topics/ materials/methods should be included in reform- oriented probability and statistics courses to improve overall student understanding of the subject. Outcome: While the project is on-going, preliminary results indicate that extensive inclusion of real-world examples, interactive lectures, student-active lab assignments, carefully crafted web-activities, and graded homework (that is connected to the previous items) result in improved student retention and mathematical understanding.

21. NSF DUE-0126716 Preliminary Survey Results Students in the project classes were given mid-term and final Class Activities Surveys. These surveys consisted of three parts: Section One asked a series of questions concerning student’s beliefs concerning his/her understanding of specific mathematics concepts covered in the course such as sample space, conditional probability, independence of events, probability laws, etc. The student was asked to rate their understanding on a 1-5 scale(L-H). Section Two included a series of questions that asked students to rate the functioning of the class in terms of class dynamics, group dynamics, instructional strategies used, amount of technology used, and the effectiveness of the technology for conveying ideas. Section Three included open-ended questions that solicited student opinions. This survey was also given to pre-project classes in Spring 2002.

22. NSF DUE-0126716 Section One Results Table 1 shows the median student responses to Section One of the survey concerning student beliefs about their understanding of core material. (N=14 in MA 331 at ASU; N=26 in MA 2050 at MTSU) In order to test the validity of the students’ answers, students were also asked to evaluate their understanding of topics that were not covered in the course. We believe 95% of survey respondents provided valid answers concerning their beliefs about their own understanding.

23. NSF DUE-0126716 Section One Results - Table 1 5 = High Knowledge 1 = Low Knowledge

24. NSF DUE-0126716 Section One Results - Table 1 5 = High Knowledge 1 = Low Knowledge

25. NSF DUE-0126716 Section One Results - Table 1 5 = High Knowledge 1 = Low Knowledge

26. NSF DUE-0126716 Section One Results - Table 1 As might be expected, the results seem to illustrate the different content area emphases in the two courses. 5 = High Knowledge 1 = Low Knowledge

27. NSF DUE-0126716 Section Two Results - Table 2 *Due to cost and availability of Mini-tab 5 = Very Good 1 = Very Poor

28. NSF DUE-0126716 Section Three Results Table 3 indicates the responses of the MA 2050 class concerning the activities that they found to be most useful.

29. NSF DUE-0126716 Section Three Results-Table 3

30. NSF DUE-0126716 Section Three Results-Table 3

31. NSF DUE-0126716 Section Three Results-Table 3

32. NSF DUE-0126716 Section Three Results-Table 3

33. NSF DUE-0126716 Section Three Results The MA 331 (N=14) class reported the following activities related to the following concepts to be the most helpful to their learning: Basic Probability Rules Conditional Probability Hypergeometric probabilities Bayes’ Theorem Central Limit Theorem

34. NSF DUE-0126716 Further Data Acquisition During the spring term 2003, each project class will be observed repeatedly by the project evaluator. Individual videotaped teaching interviews will be conducted with selected students from each class and case studies will be developed from these interviews and the written artifacts of student work including, tests, homework, and reports.

35. NSF DUE-0126716 Dissemination Presentations at Professional Conferences In-Service Training for High School Statistics Teachers (Spring 2004) Summer Workshop for College Faculty (Summer 2004) Papers in Mathematics Education Journals Project Website: http://www.athens.edu/NSF_Prob_Stat/